# Small Data Global Well-Posedness and Scattering for the Inhomogeneous Nonlinear Schrödinger Equation in $H_{s}(R_{n})$

### JinMyong An

Kim Il Sung University, Pyongyang, Democratic People’s Republic of Korea### JinMyong Kim

Kim Il Sung University, Pyongyang, Democratic People’s Republic of Korea

## Abstract

We consider the Cauchy problem for the inhomogeneous nonlinear Schrödinger (INLS) equation

where $0<s<min{n,2n +1}$, $0<b<min{2,n−s,1+2n−2s }$ and $f(u)$ is a nonlinear function that behaves like $λ∣u∣_{σ}u$ with $λ∈C$ and $σ>0$. We prove that the Cauchy problem of the INLS equation is globally well-posed in $H_{s}(R_{n})$ if the initial data is sufficiently small and $σ_{0}<σ<σ_{s}$, where $σ_{0}=n4−2b $ and $σ_{s}=n−2s4−2b $ if $s<2n $, $σ_{s}=∞$ if $s≥2n $. Our global well-posedness result improves the one of Guzmán [Nonlinear Anal. Real World Appl. 37 (2017), 249–286] by extending the validity of $s$ and $b$. In addition, we also have the small data scattering result.

## Cite this article

JinMyong An, JinMyong Kim, Small Data Global Well-Posedness and Scattering for the Inhomogeneous Nonlinear Schrödinger Equation in $H_{s}(R_{n})$. Z. Anal. Anwend. 40 (2021), no. 4, pp. 453–475

DOI 10.4171/ZAA/1692